Paxos: intersection carries information only with invariants

A proposal pairs a ballot number with one value. Ballots are globally unique and totally ordered, often represented by a counter and proposer identity with a defined comparison. The same ballot must never be reused for a different value. A value becomes chosen when the protocol's accepting quorum has accepted that proposal. This fact can be true even if the proposer crashes before reporting success, so a later proposer must preserve possibly chosen history rather than requiring an old client acknowledgement as evidence.

For the ordinary majority version, any preparing quorum intersects every accepting quorum. That shared acceptor is an information bridge only because it retains its accepted ballot and value and honors its promises. If it could forget them on restart, set intersection would remain mathematically true while the safety argument collapsed. Generalized quorum systems can use different phase-one and phase-two shapes, but their required cross-phase intersection and selection invariants must be proved; arbitrary small quorums cannot be substituted into the majority example.

A proposer begins with a fresh ballot and sends prepare requests. An acceptor that can promise that ballot records a promise not to accept lower ballots and reports its highest accepted proposal, if any. The promise and relevant records must survive a crash before the response is sent. Once the proposer has a qualifying set of responses, it chooses the value attached to the highest accepted ballot among them. It may choose a new proposed value only if none reports an accepted proposal for this slot.

The selection rule is deliberately conservative. An accepted proposal reported by one responder might never have reached a quorum, yet another earlier value might already have been chosen. The induction behind Paxos ensures the highest reported accepted proposal carries the value that must be preserved when a prior choice exists. Do not replace the rule with the most common reported value, the newest client request, or the proposer's preferred value. A quorum is not an informal opinion poll: ballots determine which reported history controls the next proposal.

In phase two the proposer asks acceptors to accept its selected ballot-value pair. An acceptor must reject a request below a promise it has already made. Accepted state is persisted before a positive acceptance response is allowed to count. Reaching the required accepting quorum chooses the value, while notifying a learner or client is a later observation. Competing proposers can repeatedly preempt each other with larger ballots, preserving safety while preventing progress. Stable proposer selection or suitable backoff addresses progress without weakening the selection rule.

The reliability lab can display a majority threshold, not this two-phase history. Choose Reset, explicitly set n=5 and k=2, predict three available acceptors, then Play or Step. Reset and explicitly set n=7 and k=3 for another four-voter majority. The lab does not reveal quorum membership identities, promised ballots, accepted values, or persistence. Use the numerical readout alongside a hand-written set intersection and response table; it cannot demonstrate that the highest-accepted-value rule was followed.